Washer Drain Vent

I'm placing a laundry area under the stairs of my 3 fam. The house waste drain is about 4-5 feet away. The stack is about 20 feet away. For venting of the DWV I was thinking of taping into the stack vent in the attic, running it down to the basement along side the chimney and then horizontally along the basement ceiling (slightly pitched). Any opinions if this will help the draining of the washer and meet "code".